Hypercubic billiard word

Billiard scales are one possible generalization of MOS scales to higher arities. Considered as infinite words, a standard term for them is cutting words. They can be visualized by considering a point particle (a "billiard ball") bouncing off walls in a closed cubic room. The ratio between the numbers of the step sizes (associated with the direction of the billiard ball) may be rational (resulting in a periodic scale) or irrational (resulting in an aperiodic scale). In the binary case with irrational slope, billiard scales correspond to Sturmian words, which are aperiodic "MOS" (i.e. strict variety 2) scales.

Mathematical overview

In the rational case, let w be a scale word with signature a₁X₁ ... a_rX_r (i.e. w is a scale word with a_i-many X_i steps) and let a = (a₁, ..., a_r), which we call the velocity. We call w a rank-r billiard scale if there exists a vector b ∈ R^r such that the line at + b has intersections with coordinate level planes x_i = k ∈ Z that spell out the scale as you move in the positive t direction along that line.

This definition is equivalent to the definition given in terms of a billiard ball in a cubic room: We first fire off the billiard ball in the direction a = (a₁, ..., a_r) given by the scale signature. For integer a_i, the particle's trajectory will be periodic, and for almost any starting point, the particle will only collide with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale of the given signature, though for arity higher than 2, this can yield rotationally inequivalent scales depending on the starting point.

Identifying opposite sides of the cubic room, thereby producing an r-torus, yields an equivalent and at times more compelling visualization: now the particle always travels with velocity a, and every time a boundary is crossed and the corresponding scale step recorded, the particle reappears on the other side instead of bouncing. Considering the lines parallel to a that do not yield a billiard scale — namely those that have a point that has multiple integer coordinates — subdivides the r-torus into finitely many regions each of which gives rise to a billiard scale.

Properties

Proofs to be added

• A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale.
• Not all billiard scales are Fokker blocks; blackdye can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.
• A billiard scale becomes a billiard scale of lower rank when one removes all instances of some subset of its step sizes. However, the converse is false.
  • That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.
• Not all billiard words of arity higher than 2 are balanced. (In binary scales, the term "(1-)balanced" becomes one characterization of the MOS property.) Ternary billiard scales that are 1-balanced are MV3.

Determining whether a scale word is a billiard scale

1. Numbered list item

Questions

1. A circular word s is d-balanced if for any k ≥ 1 and for any pair of length-k subwords w and w' of s,

[math]\displaystyle{ \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|_{x_i} \big| : x_i \text{ is a letter of }s\text{ and }k = \operatorname{len}(w) = \operatorname{len}(w') \big\} \leq d, }[/math]

where |u|_xi is the number of occurrences of the letter x_i in the word u. Is it the case that for all r ≥ 1, all r-ary billiard circular words are (r − 1)-balanced? (The answer is known to be "yes" for r = 1, 2.)

2. Must ternary billiard scales have maximum variety at most 4, the MV4 ones being exactly the ones of balance 2? Does billiardness impose a deterministic relationship between arity, maximum variety and balance?

See also

• List of ternary billiard words of length at most 31